New examples of manifolds with strictly positive curvature

Eschenburg, Jost-Hinrich

doi:10.1007/bf01389224

Cited by 147 publications

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“…These are circle quotients of HP 2 . In this case the weight matrices are of the form Ω = p = (p 1 , p 2 , p 3 ) with the admissible set Thus S(p) ∼ = SU (3)/U (1) is a biquotient similar to the examples considered by Eschenburg in [14]. The action of the group SU (2) generated by {ξ 1 , ξ 2 , ξ 3 } on N (p) ∼ = SU (3) commutes with the action of U (1).…”

Section: = Cpmentioning

confidence: 99%

Some Examples of Toric Sasaki—Einstein Manifolds

Coevering

2009

Riemannian Topology and Geometric Structures on Manifolds

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Abstract. A series of examples of toric Sasaki-Einstein 5-manifolds is constructed which first appeared in the author's Ph.D. thesis [40]. These are submanifolds of the toric 3-Sasakian 7-manifolds of C. Boyer and K. Galicki. And there is a unique toric quasi-regular Sasaki-Einstein 5-manifold associated to every simply connected toric 3-Sasakian 7-manifold. Using 3-Sasakian reduction as in [8,7] an infinite series of examples is constructed of each odd second Betti number. They are all diffeomorphic to #kM∞, where M∞ ∼ = S 2 × S 3 , for k odd. We then make use of the same framework to construct positive Ricci curvature toric Sasakian metrics on the manifolds X∞#kM∞ appearing in the classification of simply connected smooth 5-manifolds due to Smale and Barden. These manifolds are not spin, thus do not admit Sasaki-Einstein metrics. They are already known to admit toric Sasakian metrics (cf. [9]) which are not of positive Ricci curvature. We then make use of the join construction of C. Boyer and K. Galicki first appearing in [6], see also [9], to construct infinitely many toric Sasaki-Einstein manifolds with arbitrarily high second Betti number of every dimension 2m + 1 ≥ 5. This is in stark contrast to the analogous case of Fano manifolds in even dimensions. IntroductionA new series of quasi-regular Sasaki-Einstein 5-manifolds is constructed. These examples first appeared in the author's Ph.D. thesis [40]. They are toric, and arise as submanifolds of toric 3-Sasakian 7-manifolds. Applying 3-Sasakian reduction to torus actions on spheres C. Boyer, K. Galicki, et al [8] produced infinitely many toric 3-Sasakian 7-manifolds. More precisely, one has a 3-Sasakian 7-manifold S Ω for each integral weight matrix Ω satisfying some conditions to ensure smoothness. This produces infinitely many examples of each b 2 (S Ω ) ≥ 1. A result of D. Calderbank and M. Singer [12] shows that, up to finite coverings, this produces all examples of toric 3-Sasakian 7-manifolds. Associated to each S Ω is its twistor space Z, a complex contact Fano 3-fold with orbifold singularities. The action of T 2 complexifies to T 2 C = C * × C * acting on Z. Furthermore, if L is the line bundle of the complex contact structure, the action defines a pencilwhere t C is the Lie algebra of T 2 C . We determine the structure of the divisors in E. The generic X ∈ E is a toric variety with orbifold structure whose orbifold anti-canonical bundle K −1 X is positive. The total space M of the associated S 1 orbifold bundle to K X has a natural Sasaki-Einstein structure. Associated to any [3]) that M is diffeomorphic to #k(S 2 × S 3 ) where b 2 (M ) = k. We have the following: Theorem 1.1. Associated to every simply connected toric 3-Sasakian 7-manifold S is a toric quasi-regular Sasaki-. This gives an invertible correspondence. That is, given either X or M in diagram 1.1 one can recover the other spaces with their respective geometries.This gives an infinite series of quasi-regular Sasaki-Einstein structures on #m(S 2 × S 3 ) for every odd m ≥ 3.In section...

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Section: = Cpmentioning

confidence: 99%

Some Examples of Toric Sasaki—Einstein Manifolds

Coevering

2009

Riemannian Topology and Geometric Structures on Manifolds

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“…By a k×k block of Spin(9) we mean that the epimorphism Spin(9) → SO(9) maps the group H up to conjugacy to a k × k block of SO (9).…”

Section: B) G Is One Of the Other Exceptional Groups And Hmentioning

confidence: 99%

Positively curved manifolds with symmetry

Wilking

2006

Ann. Math.

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There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diffeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold (M, g) is up to homotopy given by a rank one symmetric space, provided that its isometry group Iso(M, g) is large. More precisely we prove first that if dim(Iso(M, g)) ≥ 2 dim(M ) − 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M is homogeneous. Secondly, we show that in dimensions above 18(k + 1) 2 each M is tangentially homotopically equivalent to a rank one symmetric space, where k > 0 denotes the cohomogeneity, k = dim(M/ Iso(M, g)).

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“…Note that S 1 a,b SU(3) × U(2), but its action is the same, up to an ineffective kernel, as that by S 1 3ā−c,3b−c ⊂ SU(3)× U(2), wherec = (c, c, c). In [Es1] it was shown that the Eschenburg metric on Eā ,b has positive sectional curvature if and only if one of the following holds:…”

Section: Eschenburg and Bazaikin Spacesmentioning

confidence: 99%

Symmetries of Eschenburg spaces and the Chern problem

Grove¹,

Shankar²,

Ziller³

2006

Asian Journal of Mathematics

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Dedicated to the memory of S. S. Chern.To advance our basic knowledge of manifolds with positive (sectional) curvature it is essential to search for new examples, and to get a deeper understanding of the known ones. Although any positively curved manifold can be perturbed so as to have trivial isometry group, it is natural to look for, and understand the most symmetric ones, as in the case of homogeneous spaces. In addition to the compact rank one symmetric spaces, the complete list (see [BB] Our purpose here is to begin a systematic analysis of the isometry groups of the remaining known manifolds of positive curvature, i.e., of the so-called Eschenburg spaces, E 7 [Es1, Es2] (plus one in dimension 6) and the Bazaikin spaces, B 13 [Ba1], with an emphasis on the former. Any member of E is a so-called bi-quotient of SU(3) by a circle:They contain the homogeneous Aloff-Wallach spaces A, corresponding to l i = 0, i = 1, 2, 3, as a special subfamily. Similarly, any member of B is a bi-quotient of SU (5) by Sp(2) S 1 and the Berger space, B 13 ∈ B. It was already noticed several years ago by the first and last author, that both E and B contain an infinite family E 1 respectively B 1 of cohomogeneity one, i.e., their isometry groups have 1-dimensional orbit spaces (see section 1 and [Zi]). The work in [GWZ] shows that up to diffeomorphism, these are the only manifolds from E and B of cohomogeneity one. There is a larger interesting subclass E 2 ⊂ E, corresponding to l 1 = l 2 = 0, which contains E 1 as well as A, and whose members have cohomogeneity two. We point out that E 1 ∩ A has only one member A 1,1 , the unique Aloff-Wallach space that is also a normal homogeneous space (see [Wi1]).From our results about isometry groups we get in particular Theorem A. The isometry group of any E ∈ E 2 has dimension 11, 9, 7 or 5, corresponding to the cases E =

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New examples of manifolds with strictly positive curvature

Cited by 147 publications

References 6 publications

Some Examples of Toric Sasaki—Einstein Manifolds

Some Examples of Toric Sasaki—Einstein Manifolds

Positively curved manifolds with symmetry

Symmetries of Eschenburg spaces and the Chern problem

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